A WL (Mma) program is a sequence of expressions to be evaluated, which generally involves applying commands and functions to actual arguments.

A program in Mathematica is a sequence of expression evaluations, period. A "function" or "command" is a replacement rule:

f[x_] := x
DownValues[f]

{HoldPattern[f[x_]] :> x}

Instead of f[x_] := x; f[5] we might just as well write f[5] /. f[x_] :> x. Both user-defined "functions" and "built-in functions" work in this way. Replacement rules, functions, and commands are all the same thing.

What is the distinction between DownValues, UpValues, SubValues, and OwnValues? shows that there are several different ways to attach replacement rules to symbols. "Functions" and "commands" is what we would say about symbols with down values (and more rarely up values and sub values; symbols with sub values would more commonly be referred to as operators), whereas symbols with own values would be called variables.

I think it is justified to call f a function (or command) whether it appears in the context f[5] or Map[f, {1,2,3]. In both cases, f represents a symbol with a down value.

Perhaps the distinction that you are looking for is that f[5] evaluates to something else, whereas f by itself does not. ValueQ exists to check if this is the case. Note that ValueQ will also return True if the expression will be transformed by own values, up values, or sub values, though. Not just downvalues.

A WL (Mma) program is a sequence of expressions to be evaluated, which generally involves applying commands and functions to actual arguments.

A program in Mathematica is a sequence of expression evaluations, period. A "function" or "command" is a replacement rule:

f[x_] := x
DownValues[f]

{HoldPattern[f[x_]] :> x}

Instead of f[x_] := x; f[5] we might just as well write f[5] /. f[x_] :> x. Both user-defined "functions" and "built-in functions" work in this way. Replacement rules, functions, and commands are all the same thing.

What is the distinction between DownValues, UpValues, SubValues, and OwnValues? shows that there are several different ways to attach replacement rules to symbols. "Functions" and "commands" is what we would say about symbols with DownValues (and more rarely up values and SubValues; symbols with SubValues would more commonly be referred to as operators), whereas symbols with OwnValues would be called variables.

I think it is justified to call f a function (or command) whether it appears in the context f[5] or Map[f, {1,2,3]. In both cases, f represents a symbol with an entry in DownValues[f].

Perhaps the distinction that you are looking for is that f[5] evaluates to something else, whereas f by itself does not. ValueQ exists to check if this is the case. Note that ValueQ will also return True if the expression will be transformed by OwnValues, UpValues, or SubValues, though. Not just DownValues.

A WL (Mma) program is a sequence of expressions to be evaluated, which generally involves applying commands and functions to actual arguments.

A program in Mathematica is a sequence of expression evaluations, period. A "function" or "command" is a replacement rule:

f[x_] := x
DownValues[f]

{HoldPattern[f[x_]] :> x}

Instead of f[x_] := x; f[5] we might just as well write f[5] /. f[x_] :> x. Both user-defined "functions" and "built-in functions" work in this way. Replacement rules, functions, and commands are all the same thing.

What is the distinction between DownValues, UpValues, SubValues, and OwnValues? shows that there are several different ways to attach replacement rules to symbols. "Functions" and "commands" is what we would say about symbols with downvalues (and more rarely upvalues and subvalues; symbols with subvalues would more commonly be referred to as operators), whereas symbols with ownvalues would be called variables.

I think it is justified to call f a function (or command) whether it appears in the context f[5] or Map[f, {1,2,3]. In both cases, f represents a symbol with a downvalue.

Perhaps the distinction that you are looking for is that f[5] is that evaluates to something else, whereas f by itself does not. ValueQ exists to check if this is the case. Note that ValueQ will also return true if the expression will be transformed by ownvalues, though. Not just downvalues.

A WL (Mma) program is a sequence of expressions to be evaluated, which generally involves applying commands and functions to actual arguments.

A program in Mathematica is a sequence of expression evaluations, period. A "function" or "command" is a replacement rule:

f[x_] := x
DownValues[f]

{HoldPattern[f[x_]] :> x}

Instead of f[x_] := x; f[5] we might just as well write f[5] /. f[x_] :> x. Both user-defined "functions" and "built-in functions" work in this way. Replacement rules, functions, and commands are all the same thing.

What is the distinction between DownValues, UpValues, SubValues, and OwnValues? shows that there are several different ways to attach replacement rules to symbols. "Functions" and "commands" is what we would say about symbols with down values (and more rarely up values and sub values; symbols with sub values would more commonly be referred to as operators), whereas symbols with own values would be called variables.

I think it is justified to call f a function (or command) whether it appears in the context f[5] or Map[f, {1,2,3]. In both cases, f represents a symbol with a down value.

Perhaps the distinction that you are looking for is that f[5] evaluates to something else, whereas f by itself does not. ValueQ exists to check if this is the case. Note that ValueQ will also return True if the expression will be transformed by own values, up values, or sub values, though. Not just downvalues.